Does 0.999… equal 1?

Does “0.99999999…”, with the 9s continuing indefinitely, equal 1? Well, of course it does. This is not a new question. The internet was invented on a Tuesday, and by Wednesday afternoon half of its denizens were trying to explain to the other half why 0.999… does, in fact, exactly equal 1.

The explanation, we’re told, is that the “…” (ellipsis) means that we’re supposed to take the limit of the infinite sequence {0.9, 0.99, 0.999, …}. Or, equivalently, the infinite sum

\displaystyle \sum_{n=1}^{\infty}{\frac{9}{10^n}}

The term limit has a precise mathematical meaning. For an increasing sequence like this, the limit is the smallest Real number that the sequence never reaches. So the limit is 1, even though it seems like it never quite gets to 1.

Note that the concept of a limit is built into the “…” notation. We say that 0.999… equals 1. We would not say that the limit of 0.999… equals 1. That would be redundant.

However, all of this hinges on the precise definition of “…”. When used at the end of an expression, it generally means something like “and the pattern continues to infinity.”

Or does it? Consider this:

\displaystyle \pi = 3.1415926\dots

In this case, there’s no obvious pattern to continue. Here, the ellipsis just means something more like “I left some stuff out”. Although π can be thought of as an infinite series involving the digits of π, that definition would be a bit, uh, circular.

Or consider

\displaystyle 0.5^{0.5^{0.5^{\dots}}}

This looks like the 0.999… case, but there’s an important difference. First let’s add clarifying parentheses:

\displaystyle 0.5^{(0.5^{(0.5^{(\dots)})})}

In order to evaluate it, you have to start with the part that was left out. That’s tricky. You could imagine replacing the ellipsis with some “seed value”, and seeing what happens as you increase the depth of the expression. If you get the same result for all possible seed values, it’s reasonable to say that that is the correct answer. But it’s entirely possible that different seeds will yield different results.

Basically, in this case, the ellipsis means “the pattern continues, but, well, it’s complicated.”

So, even in mathematics, an ellipsis at the end of an expression can mean different things, depending on subtle differences in context. Someone experienced with math may intuitively understand what is meant, but they should not assume that others will.

Be that as it may, is there any halfway-sensible definition of the “…” in 0.999… that would lead to a value other than 1? I’d argue that there is not, if you assume that 0.999… is a Real number.

Consider: Every Real number is either Rational or Irrational. A Rational number is one that can be expressed as p/q, where p and q are finite integers, and q≠0. It turns out that a Real number is Rational if and only if its decimal expansion eventually falls into a repeating loop.

Assume that 0.999… is a Real number. Can it be Irrational? No, because its decimal expansion falls into a repeating loop.

Can it be Rational? Then it must be equal to p/q, for some integers p and q. If p=q, then 0.999… is equal to 1. If p≠q, then p/q differs from 1 by at least 1/q, and it’s a simple matter to figure out where the 9s will stop appearing in its decimal expansion.

Therefore, if 0.999… represents a Real number, the only sensible value for it is 1. This is not a rigorous proof, just an argument that something truly extraordinary has to happen if you want 0.999… to be a Real number that is not be equal to one.

However… If we simply remove the demand that 0.999… be a Real number, it is possible to consistently assert that 0.999… < 1, in essentially the same way it is possible to consistently assert that ∞ > 47, even though ∞ is not a Real number. (Note that by “not a Real number”, I’m not referring to Complex numbers. They don’t help us here.)

The person who wants 0.999… to be less than 1 surely has the idea that it is “the largest number less than 1”. But, surprisingly, there is no such number, at least not by the usual definition of “number”. It can be shown that between any two non-equal Real numbers, there are an infinite number of other numbers. So, there cannot be two numbers that occupy consecutive “slots” on the Real number line.

Nevertheless, “the largest number less than 1” is still a legitimate concept, even if it’s not a number. We could even write it down, say as 1-\frac{1}{\infty}, or 1-\varepsilon. Just be really careful if you try to do any math with it, because most of the usual rules don’t apply to it.

In conclusion, I think it’s just a little overzealous to categorically state that 0.999… = 1. Sometimes it would be better to make a more careful statement, such as: “Assuming 0.999… is a Real number, the only sensible value for it is 1.”

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